Question: Let a, b be the roots of x<sup>2</sup> –3x + p = 0 and let c, d be the roots of x<sup>2</sup> –12x +...

Question

Let a, b be the roots of x² –3x + p = 0 and let c, d be the roots of x² –12x + q = 0, where a, b, c, d form an increasing G.P. Then the ratio of q + p : q – p is equal to

Answer 1

Solution

Let b = ar, c = ar², d = ar³

a + b = 3 Ž a + ar = 3 …(1)

c + d = 12 Ž ar² + ar³ = 12 … (2)

Ž r² (a + ar) = 12 Ž r² = 4 Ž r = ±2

Increasing G.P. so r = 2 then a = 1 for (1)

a = 1, b = 2, c = 4 , d = 8

p = 1 × 2 q = 4 × 8 = 32

$\frac { q + p } { q - p }$ = $\frac { 32 + 2 } { 32 - 2 }$ = $\frac { 34 } { 30 }$ = $\frac { 17 } { 15 }$